#10102224

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(2,7)(4,9)(8,13)(12,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(35,37)(40,42)(43,44)(47,48);(2,7)(3,10)(8,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009420);